Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16149 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,9)(3,13,21,24,14,23,16,8)(4,10,18,25,20,28,19,27)(15,30,17,31,29,26,32,22);(1,3,6,21,7,14,5,16)(2,8,9,23,12,24,11,13)(4,17,18,29,20,32,19,15)(10,26,27,31,28,30,25,22);(2,9)(3,14)(5,6)(8,13)(10,25)(11,12)(15,29)(18,19)(22,26)(23,24)(27,28)(30,31);(1,4)(2,10,12,28)(3,15,14,29)(5,18)(6,19)(7,20)(8,22)(9,25,11,27)(13,26)(16,17,21,32)(23,30)(24,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251689, #72251692, #72251703, … |
| maximal normal subgroup of | #72251689, #72251692, #72251703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361039);